Time Dependent, Modal Reduced-Order Model

) to save the reduced-order model to file as COMSOL Reduced Model file (MPHROM-file). Note that export is only possible if all defined outputs are linear quantities.

The window for a node contains the following sections:

Here you specify whether the reduced-order model should have stateless or stateful interface. Choose (the default) or from the list. In the former case, the reduced-order model acts as a black box that uses an internal solver and does not expose its state. In the latter case, the reduced-order model exposes a set of reduced-order equations and state DOFs to the solver used for the calling model, and you can choose whether to solve for the reduced-order model in the same way as for a physics interface.

If you choose , also specify the value to be used when evaluating outputs. It is by default set to t, which means that the reduced-order model will be evaluated for the same time as in the calling model in which it is used.

If you choose , from the list, choose (the default) or . When the setting is , the study controls the equation form for the reduced-order model. It then appears as , for example, in the section in the study step’s Settings window.

The list in this section contains the reduced-order model input quantities that were set as active in the corresponding node’s section when the reduced-order model was created. For each model control input in the column, specify a corresponding to be used when evaluating outputs from the reduced-order model, or when using the stateful interface, when solving reduced-order equations. The default expression is the name of the corresponding reduced-order model input variable. The effect of this is that the reduced-order model will be evaluated for the same value of the input as is seen by the calling model in which it is used. This is usually the desired behavior.

The table in this section specifies constraints for the modal reduced-order model. Each row corresponds to a constraint mode in the basis, and also to a potential constraint on the reduced-order model.

The Modal Reduction solver classifies constraints in the unreduced model when training the reduced-order model according to whether they depend on a reduced-order model input or not. Constraints depending on more than one reduced-order model input are considered an error. Input-dependent constraints are then compared to the constraint modes. If a constraint can be satisfied by a single constraint mode, then that mode is considered to be input-controlled. Constraints that can be satisfied by more than one constraint mode are considered an error during training. Input-controlled modes are treated differently from other modes. There are five columns in the table: , , , , and .

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The column allows for specifying the description to be used for the variable in results analysis and menus. By default it will be X, where X is the number of the constraint mode.

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The column determines whether the corresponding constraint mode will be constrained. When the constraint mode is input controlled, the check box is selected by default; otherwise, it is unchecked.

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The column contains the value to which the constraint mode state will be constrained, if the constraint is active. If the mode is input controlled, then the field will contain the name of the variable representing the expression specified for this input in the section, and the field will not be editable. Otherwise, its default value is 0.

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The column contains a variable name identifying the reduced-order model state associated with each constraint mode. The variable is defined as <rom>.state(<index>). This column is only available for reduced-order models with a stateful interface.

The constraints defined by the settings in the table are applied in a symmetric way directly on the constraint mode states. This has a few important implications:

The symmetric application of constraints means that the equations of any dependent variables used in the column of an active constraint will get a reaction term contribution. This may or may not be the desired behavior.

This section provides an overview of output quantities that the reduced-order model makes available for evaluation, both during solution and in postprocessing. These outputs are defined in the corresponding node’s section. The column shows the variable names defined by the reduced-order model.

When the check box is selected, the reduced-order model also declares the dependent variable names entered in the column. When the reduced-order model is part of another model, these dependent variables can be assigned the value of the corresponding output variable each time a solution is stored. If the reduced-order model has a stateless interface, this behavior is controlled by the setting in the section of each study step where the reduced-order model is used. If the reduced-order model has a stateful interface, this behavior is controlled by whether the reduced-order model is solved for in a given study step. Apart from the setting to solve for a reduced-order model with a stateful interface, these settings are only available for reduced-order models that have outputs.

This section displays a list of dependent variable fields which this reduced-order model can reconstruct. When the reduced-order model is used in a calling model, it defines an operator <rom_name>.eval(<expr>), where <rom_name> is the of the reduced-order model and <expr> is an arbitrary expression.

	Field reconstruction is not possible in a Reduced Model node that uses reduced-order model data saved in a COMSOL Reduced Model file and then imported. The original model data needed is then not available.

Use the to scale the constant part of the right-hand side of the unreduced model. This in practice scales all loads and source terms with the specified factor. Inhomogeneous constraints are not affected.

If the reduced-order model has a stateless interface, specify a for the BDF time-stepping method used in the internal solver. This setting is only available when is selected from the list in the section above.

This section contains information about the reduced-order model: Which study it was created from, which solution data that contains the reduced model data, the model reduction method, the unreduced study type, and some general information about the model reduction.

Under , all matrices that the reduced model solution includes are listed. The listed matrices can be accessed using the COMSOL API.

Under , all vectors that the reduced model solution includes are listed. The listed vectors can be accessed using the COMSOL API.

The build log contains output from the reduced-order model solution process, similar to other solver logs.

The matrices used to formulate the first-order and second-order state-space systems are generated by building the Time Dependent, Modal Reduced-Order Model.

If the second derivatives are zero, and Brdot (the time-derivative input matrix) is zero, let Kud denote the stiffness matrix K times Ud, which is a particular solution satisfying the constraints. Then, the state-space equations can be written in this first-order form:

The initial condition becomes

where B0r is the initial value input matrix.

The linearized output is computed by

where Y0 is the output bias vector.

With eliminated constrained modes, the state-space equations can be written in this first-order form:

The initial condition, with B0r = 0, becomes

The linearized output, with Y0 = 0, is then computed by

The eliminated MAxe and Cxe vectors are available for output (see the table below).

A second-order formulation can be rewritten on this form by adding states corresponding to

The following matrices and vectors are available with the first-order formulation:

Table 5-20: Matrices and Vectors available with the First-Order Formulation
Name	Description	Definition for the First-Order Case	Definition for the Second-Order Case
Mc	Mass matrix	Dr	[O Er; Ir 0]
MA	Stiffness matrix	-Kr	[-Kr -Dr; 0 Ir]
MB	Input matrix	-Br	[-Br; 0]
x0	Initial value vector	U0	[U0; Udot0]
C	Output matrix	Cr	[Cr 0]
D	Input feedback matrix	F	F
MAxe	Eliminated constraint vector	-Kr_e*x_e	[-Kr_e*x_e 0; 0 Ir_r]
Cxe	Vector representing sub-output matrix times constrained states	Cr_e*x_e	Cr_e*x_e

Here, Ir in the second-order definition represents the identity matrix.

To access these matrices and vectors, right-click , choose , and in the settings for the node, choose from the list. The list in the section then includes all matrices and vectors of interest.